Sets and Relations
Complex Numbers 33
Sequences and Series
Locus and Straight Line
Measures of Dispersion
Bivariate Frequency Distribution and Chi Square Statistic
Permutations and Combinations
- Introduction of Permutations and Combinations
- Fundamental Principles of Counting
- Concept of Addition Principle
- Concept of Multiplication Principle
- Concept of Factorial Function
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When All Objects Are Not Distinct
- Circular Permutations
- Properties of Permutations
- Properties of Combinations
The limiting process respects addition, subtraction, multiplication and division as long as the limits and functions under consideration are well defined. In fact, below we formalise these as a theorem without proof.
Let f and g be two functions such that both `lim_(x -> a)` f(x) and `lim_(x -> a)` g(x) exist.
(i) Limit of sum of two functions is sum of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) + g(x)]` = `lim_(x -> a) f(x) + lim _(x -> a) g(x)`.
(ii) Limit of difference of two functions is difference of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) -g(x)] = lim_(x -> a) f(x) -lim _(x -> a) g(x)`.
(iii) Limit of product of two functions is product of the limits of the functions, i.e.,
`lim_(x -> a) [f(x) . g(x)] = lim_(x -> a) f(x) . lim _(x -> a) g(x)`.
(iv) Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero), i.e.,
`lim_(x -> a) f(x)/g(x) =(lim_(x->a) f(x))/(lim_(x-> a ) g(x))`
Shaalaa.com | Algebra of limits
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